IS/ISO 11843-2 : 2000 Hkkjrh; ekud lalwpu l{kerk Hkkx 2 js[kh; va'k'kks 'k'kks/ku dsl dh i)fr Indian Standard CAPABILITY OF DETECTION PART 2 METHODOLOGY IN THE LINEAR CALIBRATION CASE ICS 03.120.03; 17.020 © BIS 2010 B U R E AU O F I N D I A N S TA N DA R D S MANAK BHAVAN, 9 BAHADUR SHAH ZAFAR MARG NEW DELHI 110002 January 2010 Price Group 9 Statistical Methods for Quality and Reliability Sectional Committee, MSD 3 NATIONAL FOREWORD This Indian Standard (Part 2) which is identical with ISO 11843-2 : 2000 `Capability of detection -- Part 2: Methodology in the linear calibration case' issued by the International Organization for Standarization (ISO) was adopted by the Bureau of Indian Standards on the recommendation of the Statistical Methods for Quality and Reliability Sectional Committee and approval of the Management and Systems Division Council. An ideal requirement for the capability of detection with respect to a selected state variable would be that the actual state of every observed system can be classified with certainty as either equal to or different from its basic state. However, due to systematic and random distortions, this ideal requirement cannot be satisfied because: -- in reality all reference states, including the basic state, are never known in terms of the state variable. Hence, all states can only be correctly characterized in terms of differences from basic state, that is, in terms of the net state variable. -- in practice, reference states are very often assumed to be known with respect to the state variable. In other words, the value of the state variable for the basic state is set to zero; for instance in analytical chemistry, the unknown concentration or the amount of analyte in the blank material usually is assumed to be zero and values of the net concentration or amount are reported in terms of supposed concentrations or amounts. In chemical trace analysis especially, it is only possible to estimate concentration or amount differences with respect to available blank material. In order to prevent erroneous decisions, it is generally recommended to report differences from the basic state only, that is, data in terms of the net state variable. -- the calibrations and the processes of sampling and sample preparation add random variation to the measurement results. In this standard, the following two requirements were chosen: -- the probability is of detecting (erroneously) that a system is not in the basic state when it is in the basic state; -- the probability is of (erroneously) not detecting that a system, for which the value of the net state variable is equal to the minimum detectable value (xd), is not in the basic state. The text of ISO Standard has been approved as suitable for publication as an Indian Standard without deviations. Certain conventions are, however, not identical to those used in Indian Standards. Attention is particularly drawn to the following: a) Wherever the words `International Standard' appear referring to this standard, they should be read as `Indian Standard'. b) Comma (,) has been used as a decimal marker in the International Standard while in Indian Standards, the current practice is to use a point (.) as the decimal marker. (Continued on third cover) IS/ISO 11843-2 : 2000 Indian Standard CAPABILITY OF DETECTION PART 2 METHODOLOGY IN THE LINEAR CALIBRATION CASE 1 Scope This part of ISO 11843 specifies basic methods to: ¾ ¾ design experiments for the estimation of the critical value of the net state variable, the critical value of the response variable and the minimum detectable value of the net state variable, estimate these characteristics from experimental data for the cases in which the calibration function is linear and the standard deviation is either constant or linearly related to the net state variable. The methods described in this part of ISO 11843 are applicable to various situations such as checking the existence of a certain substance in a material, the emission of energy from samples or plants, or the geometric change in static systems under distortion. Critical values can be derived from an actual measurement series so as to assess the unknown states of systems included in the series, whereas the minimum detectable value of the net state variable as a characteristic of the measurement method serves for the selection of appropriate measurement processes. In order to characterize a measurement process, a laboratory or the measurement method, the minimum detectable value can be stated if appropriate data are available for each relevant level, i.e. a measurement series, a measurement process, a laboratory or a measurement method. The minimum detectable values may be different for a measurement series, a measurement process, a laboratory or the measurement method. ISO 11843 is applicable to quantities measured on scales that are fundamentally continuous. It is applicable to measurement processes and types of measurement equipment where the functional relationship between the expected value of the response variable and the value of the state variable is described by a calibration function. If the response variable or the state variable is a vectorial quantity the methods of ISO 11843 are applicable separately to the components of the vectors or functions of the components. 2 Normative references The following normative documents contain provisions which, through reference in this text, constitute provisions of this part of ISO 11843. For dated references, subsequent amendments to, or revisions of, any of these publications do not apply. However, parties to agreements based on this part of ISO 11843 are encouraged to investigate the possibility of applying the most recent editions of the normative documents indicated below. For undated references, the latest edition of the normative document referred to applies. Members of ISO and IEC maintain registers of currently valid International Standards. ISO 3534-1:1993, Statistics -- Vocabulary and symbols -- Part 1: Probability and general statistical terms. ISO 3534-2:1993, Statistics -- Vocabulary and symbols -- Part 2: Statistical quality control. ISO 3534-3:1999, Statistics -- Vocabulary and symbols -- Part 3: Design of experiments. 1 IS/ISO 11843-2 : 2000 ISO 11095:1996, Linear calibration using reference materials. ISO 11843-1:1997, Capability of detection -- Part 1: Terms and definitions. ISO Guide 30:1992, Terms and definitions used in connection with reference materials. 3 Terms and definitions For the purposes of this part of ISO 11843, the terms and definitions of ISO 3534 (all parts), ISO Guide 30, ISO 11095 and ISO 11843-1 apply. 4 4.1 Experimental design General The procedure for determining values of an unknown actual state includes sampling, preparation and the measurement itself. As every step of this procedure may produce distortion, it is essential to apply the same procedure for characterizing, for use in the preparation and determination of the values of the unknown actual state, for all reference states and for the basic state used for calibration. For the purpose of determining differences between the values characterizing one or more unknown actual states and the basic state, it is necessary to choose an experimental design suited for comparison. The experimental units of such an experiment are obtained from the actual states to be measured and all reference states used for calibration. An ideal design would keep constant all factors known to influence the outcome and control of unknown factors by providing a randomized order to prepare and perform the measurements. In reality it may be difficult to proceed in such a way, as the preparations and determination of the values of the states involved are performed consecutively over a period of time. However, in order to detect major biases changing with time, it is strongly recommended to perform one half of the calibration before and one half after the measurement of the unknown states. However, this is only possible if the size of the measurement series is known in advance and if there is sufficient time to follow this approach. If it is not possible to control all influencing factors, conditional statements containing all unproven assumptions shall be presented. Many measurement methods require a chemical or physical treatment of the sample prior to the measurement itself. Both of these steps of the measurement procedure add variation to the measurement results. If it is required to repeat measurements the repetition consists in a full repetition of the preparation and the measurement. However, in many situations the measurement procedure is not repeated fully, in particular not all of the preparational steps are repeated for each measurement; see note in 5.2.1. 4.2 Choice of reference states The range of values of the net state variable spanned by the reference states should include ¾ ¾ the value zero of the net state variable, i.e. in analytical chemistry a sample of the blank material, and at least one value close to that suggested by a priori information on the minimum detectable value; if this requirement is not fulfilled, the calibration experiment should be repeated with other values of the net state variable, as appropriate. The reference states should be chosen so that the values of the net state variable (including log-scaled values) are approximately equidistant in the range between the smallest and largest value. In cases in which the reference states are represented by preparations of reference materials their composition should be as close as possible to the composition of the material to be measured. 2 IS/ISO 11843-2 : 2000 4.3 Choice of the number of reference states, I, and the (numbers of) replications of procedure, J, K and L The choice of reference states, number of preparations and replicate measurements shall be as follows: ¾ ¾ ¾ ¾ the number of reference states I used in the calibration experiment shall be at least 3; however, I = 5 is recommended; the number of preparations for each reference state J (including the basic state) should be identical; at least two preparations (J = 2) are recommended; the number of preparations for the actual state K should be identical to the number J of preparations for each reference state; the number of repeated measurements performed per preparation L shall be identical; at least two repeated measurements (L = 2) are recommended. NOTE The formulae for the critical values and the minimum detectable value in clause 5 are only valid under the assumption that the number of repeated measurements per preparation is identical for all measurements of reference states and actual states. As the variations and cost due to the preparation usually will be much higher than those due to the measurement, the optimal choice of J, K and L may be derived from an optimization of constraints regarding variation and costs. 5 The critical values yc and xc and the minimum detectable value xd of a measurement series 5.1 Basic assumptions The following procedures for the computation of the critical values and the minimum detectable value are based on the assumptions of ISO 11095. The methods of ISO 11095 are used with one generalization; see 5.3. Basic assumptions of ISO 11095 are that ¾ ¾ ¾ the calibration function is linear, measurements of the response variable of all preparations and reference states are assumed to be independent and normally distributed with standard deviation referred to as "residual standard deviation", the residual standard deviation is either a constant, i.e. it does not depend on the values of the net state variable [case 1], or it forms a linear function of the values of the net state variable [case 2]. The decision regarding the applicability of this part of ISO 11843 and the choice of one of these two cases should be based on prior knowledge and a visual examination of the data. 3 IS/ISO 11843-2 : 2000 5.2 5.2.1 Case 1 -- Constant standard deviation Model The following model is based on assumptions of linearity of the calibration function and of constant standard deviation and is given by: Yi j = a + bx i + A i j where (1) xi Ai j is the symbol for the net state variable in state i; are random variables which describe the random component of sampling, preparation and measurement error. It is assumed that the A i j are independent and normally distributed with expectation zero and the theoretical residual standard deviation I : A i j ~ N 0 ; I 2 . Therefore, values Yi j of the response variable are random variables e j with the expectation E Yi j = a + bxi and the variance V Yi j = I ², not depending on xi . NOTE In the cases in which J samples are prepared for measurement and each of them is measured L times so that J×L measurements are performed altogether for reference state i, then Y i j refers to the average of the L measurements obtained on the prepared sample. d i d i 5.2.2 Estimation of the calibration function and the residual standard deviation In accordance with ISO 11095, estimates (see note) for a, b and I 2 are given by: = b å å ( xi - x )( yi j - y ) i =1 j -1 I J s xx (2) (3) = y - bx a 2 = I 1 I×J -2 ij - bx å å e yi j - a i =1 j =1 I J 2 (4) The symbols used here and elsewhere in this part of ISO 11843 are defined in annex A. NOTE Estimates are denoted by a symbol ^ to differentiate them from the parameters themselves which are unknown. 5.2.3 Computation of critical values The critical value of the response variable is given by: yc = a + t0,95 (n )s x2 1 1 + + K I × J s xx (5) 4 IS/ISO 11843-2 : 2000 The critical value of the net state variable is given by: x c = t0,95 (n ) s b x2 1 1 + + K I × J s xx (6) t0,95 n af is the 95 %-quantile of the t-distribution with n = I × J - 2 degrees of freedom. The derivation of these formulae is given in annex B. 5.2.4 Computation of the minimum detectable value The minimum detectable value is given by: xd = d where d = n ; a ; b is the value of the noncentrality parameter determined in such a way that a random variable following the noncentral t-distribution with n = I × J - 2 degrees of freedom and the noncentrality parameter d , T n ; d , satisfies the equation: 1-= s b x2 1 1 + + K I × J s xx (7) b g b g P T bn ; d g u t an f = b where t1-= (n ) is the (1-a )-quantile of the t-distribution with n degrees of freedom. The derivation of this formula is given in annex B. For a = b and n > 3, a good approximation for d is given by d (n ; a ; b ) » 2t 1-= (n ) (8) if n = 4 and a = b = 0,05, the relative error of this approximation is 5 %; t1-=(n ) is the (1-a )-quantile of the t-distribution with n = I×J - 2 degrees of freedom. Table 1 presents d (n ; a ; b ) for a = b = 0,05 and various values of n. For a = b and n > 3, xd is approximated by x d » 2t 0,95 (n ) ^ s ^ b 1 1 x2 + + = 2x c K I × J s xx (9) 5 IS/ISO 11843-2 : 2000 Table 1 -- Values of the noncentrality parameter for a = b = 0,05 and n degrees of freedom n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 d (n ; a ; b ) 5,516 4,456 4,067 3,870 3,752 3,673 3,617 3,575 3,543 3,517 3,496 3,479 3,464 3,451 3,440 3,431 3,422 n 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 d (n ; a ; b ) 3,415 3,408 3,402 3,397 3,392 3,387 3,383 3,380 3,376 3,373 3,370 3,367 3,365 3,362 3,360 3,358 3,356 n 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 d (n ; a ; b ) 3,354 3,352 3,350 3,349 3,347 3,346 3,344 3,343 3,342 3,341 3,339 3,338 3,337 3,336 3,335 5.3 5.3.1 Case 2 -- Standard deviation linearly dependent on the net state variable Model The following model is based on the assumptions that the calibration function is linear and that the standard deviation is linearly dependent on the net state variable and is given by: Yi j = a + bx i + e i j where (10) xi, a, b and Yi j are as defined in 5.2.1 and the ei j are independent and normally distributed with expectation E(ei j) = 0 and variance: V (A i j ) = I 2 ( x i ) = c + dxi b g 2 (11) i.e., the residual standard deviation is linearly dependent on x I ( xi ) = c + dxi The parameters of the model, a, b, c and d are estimated in a two part procedure as given in 5.3.2 and 5.3.3. (12) 5.3.2 Estimation of the linear relationship between the residual standard deviation and the net state variable The parameters c and d are estimated by a linear regression analysis with the standard deviations: si = 1 ( yi j - yi )2 J - 1 j =1 å J (13) 6 IS/ISO 11843-2 : 2000 as values of the dependent variable S and with the net state variable x as the independent variable. Since the variance V(S) is proportional to s 2, a weighted regression analysis (see references [1] and [2] of the Bibliography) has to be performed with the weights: wi = 1 I (xi ) 2 = 1 (c + dx i ) 2 (14) However, the variances s 2(xi) depend on the unknown parameters c and d that have yet to be estimated. Therefore, the following iteration procedure with weights: w qi = 1 qi )2 (s (15) 0i = si, where the si values are the empirical standard deviations. For is proposed. At the first iteration, (q = 0), I successive iterations q = 1,2, ... q x i qi = c I q + d calculate with the auxiliary values: (16) Tq +11 , = qi ; åw i =1 I I Tq +12 , = qi x i ; åw i =1 I Tq +13 , = qi xi 2 ; åw i =1 I (17) Tq +14 , = qi si ; åw i =1 I Tq +15 , = and qi xi si åw i =1 c q +1 = and q +1 = d Tq +13 , Tq +14 , - Tq +12 , Tq +15 , 2 Tq +11 , Tq +13 , - Tq +12 , (18) Tq +11 , Tq +15 , - Tq +12 , Tq +14 , 2 Tq +11 , Tq +13 , - Tq +12 , (19) This procedures converges rapidly so that the result for q = 3; 3 x ; 3 =c 3 + d I 7 IS/ISO 11843-2 : 2000 3 = d , as the final result: 3 =I ( x ), c 0 and d can be considered, with I 3 = I ( x ) ( x) = I 0 +d I 5.3.3 Estimation of the calibration function (20) The parameters a and b are estimated by a weighted linear regression analysis (see references [1] and [2] in the Bibliography) with the yi j as values of the dependent variable, xi as values of the independent variable and weights: wi = where 2 ( xi ) I with: T1 = J is the predicted value of the variance at xi according to equation (20) 1 I ( xi ) 2 ; å wi ; i =1 I I T2 = J å wi x i ; i =1 I T3 = J å wi xi2 ; i =1 T4 = J å å wi yi j ; i =1 j = l I J I J (21) T5 = J å å wi xi yi j i =1 j = l the estimates for a and b are: T T - T 2T 5 ^= 3 4 a 2 T1T 3 - T 2 T T - T2 T4 = 1 5 b 2 T1T3 - T2 5.3.4 Computation of critical values (22) (23) The critical value of the response variable is given by: yc = a + t0,95 (n ) 2 0 s x2 1 2 + + w s K T1 s xxw F GH I JK (24) 8 IS/ISO 11843-2 : 2000 and the critical value of the net state variable is given by: xc = where 2 t0,95 (n ) s 0 x2 1 2 + + w s K T1 s xxw b F GH I JK (25) x w = T2 / T1 2 s xxw = T3 - T2 / T1 (26) (27) 2 2 = I 1 I×J -2 ij - bx å å wi e yi j - a i =1 j =1 I J (28) and t0,95(n ) is the 95 %-quantile of the t-distribution with n = I×J - 2 degrees of freedom; sxxw is defined in annex A. 5.3.5 Computation of the minimum detectable value The minimum detectable value is given by: xd = 2 (xd ) x2 @ I 1 2 + + w I K T1 s xxw b F GH I JK (29) where d = d (n ; a ; b ) is the value of the noncentrality parameter as defined in 5.2.4. 2 ( x d ) depends on the value of xd yet to be calculated, xd has to be calculated iteratively. Since I ^ ( x d )1 = I ^ ( x d0 ) is computed and ( x d )0 = I 0 and results in xd 0; for the next iteration step I The iteration starts with I used in the formula for xd, resulting in xd 1,... In many cases even the first iteration step does not change the value of xd appreciably; an acceptable value for xd is obtained at the third iteration step. 6 Minimum detectable value of the measurement method The minimum detectable value obtained from a particular calibration shows the capability of the calibrated measurement process for the respective measurement series to detect the value of the net state variable of an observed actual state to be different from zero, i.e. it is the smallest value of the net state variable which can be detected with a probability of 1 - b as different from zero. This minimum detectable value differs for different calibrations. The minimum detectable values of different measurement series for ¾ ¾ ¾ a particular measurement process based on the same type of measurement process, a type of measurement process based on the same measurement method, or a measurement method can be interpreted as realizations of a random variable for which the parameters of the probability distribution can be considered characteristics of the measurement process, the type of measurement process or of the measurement method, respectively. 9 IS/ISO 11843-2 : 2000 If, for a particular measurement process, m consecutive calibrations have been carried out in order to determine the minimum detectable value of the net state variable xd, the m minimum detectable values xd1, xd2, ... xdm, can be used to determine a minimum detectable value of the measurement process under the following conditions: a) b) c) the measurement process is not changed; the distribution of the values xd is unimodal and there are no outlying values xd; the experimental design (including the number of reference states, I, and the numbers of replications of procedure, J, K and L) was identical for each of the calibrations. Under these conditions the median of the values xdi, for i = 1, ..., m, is recommended as the minimum detectable value of the measurement process; if another summary statistic of the values xdi is used instead of the median, the statistic used shall be reported. If any of these conditions are violated, the minimum detectable value of the measurement process is not sufficiently well-defined and the determination of a common value shall not be attempted. If the same measurement method is applied in p laboratories and for each of them a minimum detectable value of the measurement process within the laboratory were to be determined, then under the same conditions as for the determination of the minimum detectable value of the measurement process, the median of the p minimum detectable values of the laboratories is recommended as the minimum detectable value of the measurement method; if another summary statistic of the minimum detectable values of the laboratories is used instead of the median, the statistic used shall be reported. 7 Reporting and use of results Examples of the determination of critical and minimal detectable values are given in annex C. NOTE 7.1 Critical values For decisions regarding the investigation of actual states only the critical value of the net state variable or of the response variable is to be applied. These values derived from a calibration of the measurement process are decision limits to be used to assess the unknown states of systems included in this series. Looking at consecutive calibrations of the same measurement process, the critical values may vary from one calibration to another. However, since each of the critical values is a decision limit belonging to a particular measurement series, it is meaningless to calculate overall critical values across calibrations and logically inappropriate to use these as critical values. If a value of the net state variable or of the response variable is not greater than the critical value, it can be stated that no difference can be shown between the observed actual state and the basic state. However, due to the possibility of committing an error of the second kind, this value should not be construed as demonstrating that the observed system definitely is in its basic state. Therefore, reporting such a result as "zero" or as "smaller than the minimum detectable value" is not permissible. The value (and its uncertainty) should always be reported; if it does not exceed the critical value, the comment "not detected" should be added. 7.2 Minimum detectable values The minimum detectable value derived from a particular calibration shows whether the capability of detection of the actual measurement process is sufficient for the intended purpose. If it is not, the number J, K or L may be modified. A minimum detectable value derived from a set of calibrations following the conditions mentioned in clause 6 may serve for the comparison, the choice or the judgement of different laboratories or methods, respectively. 10 IS/ISO 11843-2 : 2000 Annex A (normative) Symbols and abbreviations a intercept in the expression y = a + bx + A estimate of the intercept a slope in the expression y = a + bx + A estimate of the slope b intercept in the expression I ( x ) = c + dx for the residual standard deviation a b b c c d d E() I i = 1, ..., I J j = 1, ...., J K k = 1, ..., K L l = 1,..., L M m N estimate of the intercept c slope in the expression I ( x ) = c + dx estimate of the slope d expectation (of the random variable given in the brackets) number of reference states used in the calibration experiment identifying variable of the reference states number of preparations for each reference state identifying variable of preparations for the reference- and basic state number of preparations for the actual state identifying variable of preparations for the actual state number of repeated measurements for each preparation identifying variable of the repeated measurements per preparation multiplying factor number of consecutive calibrations number of preparations in the calibration experiment; if the number of preparations for each reference state is identical, then N = I×J, and the total number of measurements in the calibration experiment is N×L number of the iteration step empirical standard deviation for the residual standard deviation q = 0,1,2, ... s 11 IS/ISO 11843-2 : 2000 s xx = J å ( x i - x )2 i =1 I sum of squared deviations of the chosen values of the net state variable for the reference states (including the basic state) from the average weighted sum of squared deviations of the chosen values of the net state variable for the reference states (including the basic state) from the weighted average auxiliary value for the weighted linear regression analysis variance (of the random variable given in the brackets) weight at xi weight at xi in the qth iteration step net state variable, X = Z - z0 a particular value of the net state variable chosen values of the net state variable X for the reference states including the basic state critical value of the net state variable minimum detectable value of the net state variable s xxw = J å wi ( x i - x w )2 i =1 I T V() wi w qi X x x1, ..., xI xc xd x= 1 xi I i =1 å I average of the chosen values of the net state variable for the reference states (including the basic state) = x ya - a b estimated value of the net state variable for a specific actual state I x w= Y yc yijl å wi x i å wi i =1 i =1 I weighted average of the chosen values of the net state variable for the reference states (including the basic state) response variable critical value of the response variable lth measurement of the jth preparation of the ith reference state obtained values of the response variable for the kth preparation of a specific actual state in the measurement series K L y k 1 , ..., y k l ya = 1 ykl K × L k =1 l =1 åå I J average of the observed values for a specific actual state y= 1 I×J×L å å å yi jl i =1 j =1 l =1 L average of the measurement values yi jl yi j = 1 yi jl L l =1 å L average of the measurement values of the jth preparation of the ith reference state 12 IS/ISO 11843-2 : 2000 yi = 1 J×L åå J L yi jl average of the measurement values of the ith reference state j =1 l =1 y0 Z z0 a average of the K× L measurement values at x = 0 state variable value of the state variable in the basic state probability of erroneously rejecting the null hypothesis "the state under consideration is not different from the basic state with respect to the state variable" for each of the observed actual states in the measurement series for which this null hypothesis is true (probability of the error of the first kind) in the absence of specific recommendations the value a should be fixed at a = 0,05 b probability of erroneously accepting the null hypothesis "the state under consideration is not different from the basic state with respect to the state variable" for each of the observed actual states in the measurement series for which the net state variable is equal to the minimum detectable value to be determined (probability of the error of the second kind) in the absence of specific recommendations the value b should be fixed at b = 0,05 d e n non-centrality parameter of the non-central t-distribution component of the response variable measurement representing the random component of sampling, preparation and measurement errors degrees of freedom standard deviation of the difference between the average, y , and the estimated intercept, a estimate of the residual standard deviation standard deviation at xi in the qth iteration step estimate of the residual standard deviation, x = 0 I diff I qi I 0 I 13 IS/ISO 11843-2 : 2000 Annex B (informative) Derivation of formulae B.1 Case 1 -- Constant standard deviation Under the assumptions of 5.1 and in the case of constant standard deviation, estimations of the regression , are normally distributed with expectations coefficients, a and b E a =a and variances: bg ; =b E b ej V a = where I2 x I + b g FGH I 1 ×J s J KI 2 xx 2 2 =I ; V b s xx ej is the variance of the residuals of the averages of the L repeated measurements for each preparation. If the response variable is measured K×L times at the basic state z = z0 , x = 0 , the difference between the average y0 b g of the K×L values and the estimated intercept a follows a normal distribution with expectation: ^ = E y 0 - E a ^ = a - a = 0 E y0 - a and variance: ^ = V y 0 + V a ^ = V y0 - a Since y0 - a y -a U= 0 I diff I2 æ 1 x2ö +ç + ÷I K è I × J s xx ø 2 æ1 1 x2ö =ç + + ÷I è K I × J s xx ø 2 b g is normally distributed, the random variable follows the standardized normal distribution, and the inequality: y0 - a u u 0,95 I diff holds with probability 0,95. Since I 2 diff is unknown it can be estimated as: 2 I diff = F 1 + 1 + x II GH K I × J s JK 2 2 xx 14 IS/ISO 11843-2 : 2000 where 2 is the estimated residual variance of the regression analysis that shall be used instead. The random I variable y -a Tn = 0 s diff follows the t-distribution with n = I × J - 2 degrees of freedom, and the inequality: af y0 - a u t0,95 (n ) diff s or diff = a y0 u a + t0,95 (n )s + t0,95 (n )s where t0,95 (n ) x2 1 1 + + K I × J s xx is the 95 %-quantile of the t-distribution with n degrees of freedom, holds with probability 0,95. The right hand side of this inequality is the critical value of the response variable. yc = a + t0,95 (n )s x2 1 1 + + K I × J s xx and the critical value of the net state variable is xc = yc - a s = t0,95 (n ) b b x2 1 1 + + K I × J s xx Similar expressions describe these values when other quantiles of the t-distribution are appropriate. In order to determine the minimum detectable value xd of the net state variable, it is necessary to examine the diff in the case where the true value x of the net state variable is identical to the minimum I distribution of y - a b g detectable value xd of the net state variable, x = xd. It is required to detect this state with probability 1 - > , i.e: P LM y - a > t N s diff 0,95 (n ) x = xd = 1- b OP Q or P LM y - a u t N s diff 0,95 (n ) x = xd = b OP Q If x = xd, the expectation of y is: E y = a + bx d af 15 IS/ISO 11843-2 : 2000 and therefore: E y-a = bx d whereas: b g V y-a =I2 diff as for x = 0. b g LM y - a u t (n ) x = x OP N s Q L b y - a g - bx + bx u t = PM N s LM y - a - bx + bx s s ut = PM MM s s N L U +d O =PM u t0,95 (n )P MN c 2 (n ) n PQ P 0,95 d diff d d diff d d diff diff diff diff 0,95 (n ) x = xd OP Q 0,95 OP (n )P PP Q = P T (n ; d ) u t0,95 (n ) ; diff I diff independent of U follows since U = y - a - bx d / I diff follows the standardized normal distribution and I b g the distribution of c 2 (n ) n , the random variable T (n; d ) follows the noncentral t-distribution with n degrees of freedom and noncentrality parameter d; d = d n ; a; b for = = 0,05 or other appropriate value, if required is determined as the value of the noncentrality parameter of the noncentral t-distribution with n degrees of freedom that satisfies: b g P T (n; d ) u t1-= (n ) = b From: @= bx d I diff the expression: xd = @ I diff I =@ b b 1 1 x2 + + K I × J s xx for the minimum detectable value of the net state variable follows. 16 IS/ISO 11843-2 : 2000 For a prognosis, the estimates of b and s are inserted into the formula so that the minimum detectable value is given by: x d = d s b 1 1 x2 + + K I × J s xx , and the critical value of the net The critical value of the response variable yc is the sum of a and a multiple of I b . If, according to the recommendations, the values of the net state variable of the state variable is a multiple of I reference states are equidistantly spaced with the smallest value zero, a = 0,05 and either ¾ ¾ K = 1 (one preparation for the measurement of the actual state) or; K = J (number of preparations for the measurement of the actual state equal to this for the reference states); the multiplier: M = t0,95 (n ) 1 1 x2 + + K I × J s xx in the expressions for the critical values is a function of the number of reference states, I, and the number of preparations of each reference state, J, only. For some cases M is given in Table B.1. Table B.1 -- Determination of the multiplier factor, M For K = 1 I 3 3 5 5 5 J 1 2 1 2 4 I ×J 3 6 5 10 20 1+ 1 x2 + I×J s xx 1,35 1,19 1,26 1,14 1,07 For K = J t0,95 (n ) M 8,52 2,54 2,97 2,12 1,86 6,31 2,13 2,35 1,86 1,73 I 3 3 5 5 5 J 1 2 1 2 4 I ×J 3 6 5 10 20 I +1 x2 + I×J s xx 1,35 0,96 1,26 0,89 0,63 t0,95 (n ) M 8,54 2,04 2,97 1,66 1,09 6,31 2,13 2,35 1,86 1,73 17 IS/ISO 11843-2 : 2000 B.2 Case 2 -- Standard deviation linearly dependent on the net state variable Under the assumptions of 5.1 and in the case of the standard deviation being linearly dependent on the net state , are normally distributed with expectations: variable, the estimations of the regression coefficients, a and b =b Ea =a ; E b and variances: bg ej b g FGH T T T- T F T j = V eb GH T T - T = V a 3 1 3 1 1 3 2 2 2 2 II = F 1 + x II JK GH T s JK II = I JK s 2 2 w 1 xxw 2 2 xxw 2 where I2 is defined so that wi I 2 is the variance of the residuals of the averages of the L repeated measurements for preparation i. If the response variable is measured K×L times at the basic state Z = z0 , X = 0 , the difference between the average y of the K×L values and the estimated intercept a follows a normal distribution with expectation: E y-a =E y -E a =a-a=0 b g b g af bg and variance: V y-a =V y +V a = b g af bg 2 I0 x2 1 + + w I2 =I2 diff K T1 s xxw F GH I JK I2 diff is unknown, but can be estimated as follows: 2 I diff = where 2 0 2 I I x2 1 a 2 +V + w I = 0 + K K T1 s xxw bg F GH I JK 2 0 2 is the estimated residual variance of the weighted regression is taken from equation (20) and I I analysis, which shall be used instead. In analogy to case 1 the critical value of the response variable is: diff = a yc = a + t0,95 (n )s + t0,95 (n ) 2 0 s x2 1 2 + + w s K T1 s xxw F GH I JK and the critical value of the net state variable is: t0,95 (n ) s 2 s x2 1 0 2 x c = t0,95 (n ) diff = + + w s K T1 s xxw b b Similar expressions describe these values when other quantiles of the t-distribution are appropriate. F GH I JK 18 IS/ISO 11843-2 : 2000 These formulae include the case of constant standard deviation for which all the weights are equal to one, wi = 1 2 0 2. ,..., I so that T1 = I × J , x w = x , s xxw = s xx and I =I for i = 1 The minimum detectable value of the net state variable is: xd = @ I diff b where, for x = xd, I2 x = x d = V y x = xd + V a diff, xd = V y - a c h c h bg 2 w For a prognosis, the estimates of b and I 2 diff, xd , b and: = 2 I diff, xd = V y x = x d + V a c h bg 2 xd I b g + F 1 + x I I GH T s JK K 1 xxw 2 are inserted into the formula so that the minimum detectable value of the net state variable is given by: xd = 2 xd x2 @ I 1 2 + + w I K T s b 1 xxw b g F GH I JK 2 x d depends on the value of xd yet to be calculated the iterative procedure of 5.3.5 has to be used. Since I b g 19 IS/ISO 11843-2 : 2000 Annex C (informative) Examples C.1 Example 1 The mercury content, expressed in ng/g1) of plant materials, was measured by atomic absorption spectroscopy. Each sample was decomposed using a microwave (MLS-1200) technique and taken up in nitric acid / potassium dichromate solution. These solutions were examined through a Varian VGA-76 cold vapour reduction system leading to a gold-plated foil concentration system (MCA-90) prior to replicated atomic absorption measurements. In order to estimate the calibration function and to determine the capability of detection, each of six reference samples representing the blank concentration (x = 0) and the net concentrations x = 0,2 ng/g; 0,5 ng/g; 1,0 ng/g; 2,0 ng/g; 3,0 ng/g was prepared three times and each prepared sample measured once. Hence, I = 6; J = 3; L = 1. It was assumed that the assumptions of linearity of the calibration function, constant standard deviation and normal distribution of the response variable hold; a and b had been fixed in advance at a = b = 0,05 . For the determination of the concentration of mercury in the material to be analysed, two different approaches were taken into consideration: a) one measurement would be carried out (K = L = 1); or b) three samples would be prepared for measurement and each of them measured once (K = 3; L = 1) and the average ya of the observed values used as the measurement result. The results of the calibration experiment are given in Table C.1. Table C.1 -- Results of the calibration experiment for the determination of mercury content in food or drugs Reference sample i Net concentration of mercury xi ng/g Absorbance yi j 0,003 0,004 0,011 0,023 0,048 0,071 - 0,001 1 2 3 4 5 6 0 0,2 0,5 1,0 2,0 3,0 0,002 0,005 0,012 0,023 0,048 0,072 0,005 0,011 0,023 0,047 0,072 1) 1 part per billion (ppb) = 10-9 g/g = 1 ng/g. The use of ppb is deprecated. 20 IS/ISO 11843-2 : 2000 The statistical analysis yields: x = 1116 , 7 ng/g s xx = 20,425 a = 9,995 9 ´ 10 -5 = 0,023 74 b = 1109 I , 9 ´ 10 -3 Since n = N - 2 = 16; ; t0,95 (n ) = t0,95 (16) = 1746 , d (n; a ; b ) = d (16; 0,05; 0,05) = 3,440 ; d2t 0,95 (n ) = 3,492 i yc = 0,003 05 x c = 0,086 ng /g x d = 0,173 ng/g The results for the approach a) are critical value of the response variable [see equation (5)] critical value of the net concentration [see equation (6)] minimum detectable net concentration [see equation (7)] ¾ ¾ the smallest absorbance value which can be interpreted as coming from a sample with a net mercury concentration larger than the blank concentration is yc = 0,003 05 , the critical value of the response variable; the smallest net concentration of mercury in a sample which can be distinguished (with a probability of 1 - > = 0,95 ) from the blank concentration is x d = 0,173 ng/g , the minimum detectable value of the net concentration. The results for approach b) are: critical value of the response variable [see equation (5)] critical value of the net concentration [see equation (6)] minimum detectable net concentration [see equation (7)] yc = 0,002 30 x c = 0,055 ng / g x d = 0,110 ng / g C.2 Example 22) The amount of toluene in 100 ml of extracts was measured using gas chromatography interfaced with a mass spectrometric detector (GC/MS). 100 ml samples were injected into the GC/MS system. Six reference samples were used and contained toluene in known amounts in the range 4,6 pg/100 ml to 15 000 pg/100 ml. Each sample was injected and measured four times (I = 6, J = 4, L = 1, N = 24). The measurement results are given in Table C.2. 2) D.M. ROCKE and S. LORENZATO. A Two-Component Model for Measurement Error in Analytical Chemistry. Technometrics, 1995, 37, pp. 181-182. 21 IS/ISO 11843-2 : 2000 A look at the graphical representation of the measurement results shows that the relationship between the toluene amount and the response variable (peak area) is satisfactorily linear; the standard deviation of the peak area is linearly dependent on the amount of toluene. Under the additional assumption of normal distribution of the response variable the capability of detection can be determined according to 5.3. Table C.2 -- Results of the calibration experiment for the toluene amount in 100 ml extract (1) Reference sample (2) Net amount of toluene (3) Peak area (4) Empirical standard deviation (5) (6) (7) Predicted standard deviation of iteration 1 2 2i I 3 3i I i xi pg/100 ml 4,6 23 116 580 3 000 15 000 29,80 44,60 207,70 894,67 5 350,65 20 718,14 yi j 16,85 48,13 222,40 821,30 4 942,63 24 781,61 16,68 42,27 172,88 773,40 4 315,79 22 405,76 19,52 34,78 207,51 936,93 3 879,28 24 863,91 si 6,20 5,65 21,02 73,19 652,98 2 005,02 1i I 1 2 3 4 5 6 4,56 7,07 19,73 82,91 412,46 2 046,54 5,17 7,93 21,87 91,43 454,22 2 253,14 5,15 7,92 21,88 91,57 455,02 2 257,23 In the estimation procedure for c and d an iteratively reweighted linear regression analysis according to 5.3.2 is carried out which produces the following estimated linear regression functions: iteration 1: 1i = 3,933 23 + 0,136 174 xi I 2i = 4,482 84 + 0,149 911 xi I iteration 2: iteration 3: 3i = 4,462 28 + 0,150 185 xi s The corresponding predicted standard deviations are given in columns (5) to (7) of Table C.2. After the third iteration the results are stable so that the equation of iteration 3 can be used as the final result of part 1 of the estimation procedure, i.e.: ( x ) = 4,462 28 + 0,150 185 x I 0 = 4,462 28 I The parameters a and b of the calibration function are estimated by a weighted linear regression analysis according to clause 5.3.3 with the yi j of column (3) as values of the dependent variable, xi of column (2) as values of the independent variable and weights: wi = 1 I 2 b x g b4,462 28 + 0,150 185 x g i i = 1 2 This regression analysis yields: T1 = J xw å wi i =1 I = 0,223 306 = 15,566 9 22 IS/ISO 11843-2 : 2000 s xxw = 606,224 = 12,218 5 = 1,527 27 = 1,059 54 = N - 2 = 22 a b 2 I n t0,95 n af = t0,95 22 = 1,717 a f Therefore for K = 1, the following are obtained: critical value of the response variable [see equation (24)] yc = 20,82 the critical value of the net toluene amount in 100 ml of extract [see equation (25)] x c = 5,63 pg. The minimum detectable value is calculated iteratively: xd = I 0 the first value for xd For = = > = 0,05 , d n ; a; b = d 22; 0,05; 0,05 = 3,397 (see Table 1) and with I 0 x d = 6,135 2 and x d1 = 14,553 ; [see equation (29)] is xd0 = 11,139; it follows I 1 b g b g b g b g b g bx g with I xd with I 2 = 6,647 9 iteration 2 leads x d2 = 15,627 pg/100 ml and = 6,809 2 we get finally x d = x d3 = 15,967 pg/100 ml. d 3 The smallest peak area which can be interpreted as coming from a sample with a net toluene concentration larger than the blank concentration is yc = 20,82 , the critical value of the response variable. The smallest net amount of toluene in a sample of 100 ml extract which can be distinguished (with a probability of 1 - > = 0,95 ) from the blank concentration is x d = 15,97 pg/100 ml, the minimum detectable value of the net toluene concentration. 23 IS/ISO 11843-2 : 2000 Bibliography [1] [2] [3] DRAPER N.R. and SMITH H. Applied Regression Analysis. Wiley, New York, 1981. MONTGOMERY D.C. and PECK E.A. Introduction to Linear Regression Analysis. Wiley, New York, 1992. CURRIE L.A. Nomenclature in Evaluation of Analytical Methods Including Detection and Qualification Capabilities. IUPAC Recommendations 1995. Pure and Applied Chemistry, 67, 1995, pp. 1699-1723. 24 (Continued from second cover) In this adopted standard, reference appears to certain International Standards for which Indian Standards also exist. The corresponding Indian Standards which are to be substituted in their respective places are listed below along with their degree of equivalence for the edition indicated: International Standard ISO 3534-1 : 19931) Statistics -- Vocabulary and symbols -- Part 1: Probability and general statistical terms ISO 11843-1 : 1997 Capability of detection -- Par t 1: Terms and definitions Corresponding Indian Standard IS 7920 (Part 1) : 2008 Statistics -- Vocabulary and symbols: Par t 1 Probability and general statistical terms (third revision) IS/ISO 11843-1 : 1997 Capability of detection: Part 1 Terms and definitions Degree of Equivalence Technically Equivalent Identical The technical committee responsible for the preparation of this standard has reviewed the provisions of the following referred standards and has decided that they are acceptable for use in conjunction with this standard: International Standard ISO 3534-2 : 1993 ISO 3534-3 : 1999 ISO 11095 : 1996 ISO Guide 30 : 1992 Title Statistics -- Vocabulary and symbols -- Part 2: Statistical quality control Statistics -- Vocabulary and symbols -- Part 3: Design of experiments Linear calibration using reference materials Terms and definitions used in connection with reference materials Annex A of this standard is for normative reference whereas Annexes B and C are for information only. 1) Since revised in 2006. 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